2,658 research outputs found
Certified Algorithms: Worst-Case Analysis and Beyond
In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a ?-certified algorithm is also a ?-approximation algorithm - it finds a ?-approximation no matter what the input is. Second, it exactly solves ?-perturbation-resilient instances (?-perturbation-resilient instances model real-life instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbation-resilient instances, and solve optimization problems with hard constraints.
In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means. We also present some negative results
Center-based Clustering under Perturbation Stability
Clustering under most popular objective functions is NP-hard, even to
approximate well, and so unlikely to be efficiently solvable in the worst case.
Recently, Bilu and Linial \cite{Bilu09} suggested an approach aimed at
bypassing this computational barrier by using properties of instances one might
hope to hold in practice. In particular, they argue that instances in practice
should be stable to small perturbations in the metric space and give an
efficient algorithm for clustering instances of the Max-Cut problem that are
stable to perturbations of size . In addition, they conjecture that
instances stable to as little as O(1) perturbations should be solvable in
polynomial time. In this paper we prove that this conjecture is true for any
center-based clustering objective (such as -median, -means, and
-center). Specifically, we show we can efficiently find the optimal
clustering assuming only stability to factor-3 perturbations of the underlying
metric in spaces without Steiner points, and stability to factor
perturbations for general metrics. In particular, we show for such instances
that the popular Single-Linkage algorithm combined with dynamic programming
will find the optimal clustering. We also present NP-hardness results under a
weaker but related condition
Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-Means
We investigate the complexity of solving stable or perturbation-resilient
instances of k-Means and k-Median clustering in fixed dimension Euclidean
metrics (or more generally doubling metrics). The notion of stable or
perturbation resilient instances was introduced by Bilu and Linial [2010] and
Awasthi et al. [2012]. In our context we say a k-Means instance is
\alpha-stable if there is a unique OPT solution which remains unchanged if
distances are (non-uniformly) stretched by a factor of at most \alpha. Stable
clustering instances have been studied to explain why heuristics such as
Lloyd's algorithm perform well in practice. In this work we show that for any
fixed \epsilon>0, (1+\epsilon)-stable instances of k-Means in doubling metrics
can be solved in polynomial time. More precisely we show a natural multiswap
local search algorithm in fact finds the OPT solution for (1+\epsilon)-stable
instances of k-Means and k-Median in a polynomial number of iterations. We
complement this result by showing that under a plausible PCP hypothesis this is
essentially tight: that when the dimension d is part of the input, there is a
fixed \epsilon_0>0 s.t. there is not even a PTAS for (1+\epsilon_0)-stable
k-Means in R^d unless NP=RP. To do this, we consider a robust property of CSPs;
call an instance stable if there is a unique optimum solution x^* and for any
other solution x', the number of unsatisfied clauses is proportional to the
Hamming distance between x^* and x'. Dinur et al. have already shown stable
QSAT is hard to approximate for some constant Q, our hypothesis is simply that
stable QSAT with bounded variable occurrence is also hard. Given this
hypothesis, we consider "stability-preserving" reductions to prove our hardness
for stable k-Means. Such reductions seem to be more fragile than standard
L-reductions and may be of further use to demonstrate other stable optimization
problems are hard.Comment: 29 page
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
k-Center Clustering Under Perturbation Resilience
The k-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric kcenter and an O(log*(k))-approximation for the asymmetric version. Therefore to improve on these ratios, one must go beyond the worst case.
In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric k-center problems under a very natural input stability (promise) condition called alpha-perturbation resilience [Bilu Linial, 2012], which states that the optimal solution does not change under any alpha-factor perturbation to the input distances. We show that by assuming 2-perturbation resilience, the exact solution for the asymmetric k-center problem can be found in polynomial time. To our knowledge, this is the first problem that is hard to approximate to any constant factor in the worst case, yet can be optimally solved in polynomial time under perturbation resilience for a constant value of alpha. Furthermore, we prove our result is tight by showing symmetric k-center under (2-epsilon)-perturbation resilience is hard unless NP=RP.
This is the first tight result for any problem under perturbation resilience, i.e., this is the first time the exact value of alpha for which the problem switches from being NP-hard to efficiently computable has been found.
Our results illustrate a surprising relationship between symmetric and asymmetric k-center instances under perturbation resilience. Unlike approximation ratio, for which symmetric k-center is easily solved to a factor of 2 but asymmetric k-center cannot be approximated to any constant factor, both symmetric and asymmetric k-center can be solved optimally under resilience
to 2-perturbations
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